🤖 AI Summary
This paper establishes a rigorous mathematical analogy between conditional probability and geometric path lifting in the Wasserstein space. Method: It introduces the notion of submetry into the Wasserstein metric structure for the first time, proving that conditional probability corresponds bijectively to cost-preserving lifts of optimal transport plans; it further develops a “lens” framework within weighted categories to model conditional probability as structured morphisms. The approach integrates optimal transport theory, measure theory on standard Borel spaces, pseudometric geometry, and weighted category theory. Contributions: (1) A categorical characterization of the Wasserstein distance, whose value arises as the solution to an optimization problem over weighted-category morphisms; (2) A one-to-one correspondence between conditional probabilities and cost-preserving lifts; (3) A unification of optimal transport, metric geometry, and category theory, yielding an intrinsic geometric interpretation of probabilistic structures.
📝 Abstract
This paper makes mathematically precise the idea that conditional probabilities are analogous to path liftings in geometry. The idea of lifting is modelled in terms of the category-theoretic concept of a lens, which can be interpreted as a consistent choice of arrow liftings. The category we study is the one of probability measures over a given standard Borel space, with morphisms given by the couplings, or transport plans. The geometrical picture is even more apparent once we equip the arrows of the category with weights, which one can interpret as"lengths"or"costs", forming a so-called weighted category, which unifies several concepts of category theory and metric geometry. Indeed, we show that the weighted version of a lens is tightly connected to the notion of submetry in geometry. Every weighted category gives rise to a pseudo-quasimetric space via optimization over the arrows. In particular, Wasserstein spaces can be obtained from the weighted categories of probability measures and their couplings, with the weight of a coupling given by its cost. In this case, conditionals allow one to form weighted lenses, which one can interpret as"lifting transport plans, while preserving their cost".